Nuprl Lemma : union_functionality_wrt_uiff
∀[P1,P2,Q1,Q2:ℙ]. ({uiff(P1;P2)} ⇒ {uiff(Q1;Q2)} ⇒ {P1 + Q1 ⇐⇒ P2 + Q2})
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
union: left + right
Definitions unfolded in proof :
guard: {T},
or: P ∨ Q,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q
Lemmas referenced :
or_wf,
uiff_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
unionElimination,
inlFormation,
independent_isectElimination,
hypothesis,
hypothesisEquality,
inrFormation,
cut,
introduction,
extract_by_obid,
isectElimination,
Error :inhabitedIsType,
Error :universeIsType,
universeEquality
Latex:
\mforall{}[P1,P2,Q1,Q2:\mBbbP{}]. (\{uiff(P1;P2)\} {}\mRightarrow{} \{uiff(Q1;Q2)\} {}\mRightarrow{} \{P1 + Q1 \mLeftarrow{}{}\mRightarrow{} P2 + Q2\})
Date html generated:
2019_06_20-AM-11_17_22
Last ObjectModification:
2018_09_26-AM-10_24_50
Theory : core_2
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